Part of my Calculus procedure has been taking some benchmark data on my kids throughout the years. Other than improving student attitudes about Calculus, the second big priority is making sure students have an informed opinion about how they might do on the AP Exam. Kids are always free to do what they want, but I want to make sure if they're going to spend the money on that thing that they have a shot. Our results have been creeping upwards, and we are poised for a breakthrough, at least I hope so.

My data collection schemes have been problematic though. I think I've been a little too aggressive, giving questions that students probably aren't ready for in December. With the significant hurricane delay, we weren't even ready for what I've tried in the past, so I needed a new scheme. And with 75 students in AB, I needed something that'd be efficient to process so students could get feedback quickly.

Design

I wanted to test three things: Fundamentals (trig values, limits, continuity), Interpretation (curve sketching), and Skills (derivatives rules, Riemann sums). Roughly 12-20 items per section. I wasn't going to belabor any skills, if you can do it once you can do it ten times I figure. I sketched out what I wanted in each section:

To gather all the information, I was going to use Desmos Activity Builder. I didn't want to juggle a lot of papers, and I wanted a better idea of what items were causing problems. With previous benchmarks I had a vague idea of which questions didn't go well, this time I wanted to know for sure.

I included a mix of items: entering answers, typing short answers, multiple choice, plucking data off a graph, sketching on top of a graph, and some screens where problems were presented that students would complete on little cards they'd hand in. I wanted to assess their ability to determine a limit/derivative without making math entry fluency a limiting factor.

Implementation

I initially planned 3 versions with 6 codes, but the reality of sifting through all the dashboards made me reconsider. I settled on 3 versions with 3 codes, randomly distributed among my class periods. There were 44 screens total, and 25 kids on each code.

The "version numbers" are just arbitrary hexadecimal numbers (go ahead, convert them, see how dumb I am) designed to obscure the number of versions. I was giving this to a lot of kids all day long over multiple days, I knew they were going to discuss it, but I wanted to make it a tiny bit less likely that they could figure out who they were sharing versions with.

Again for data collection simplicity, kids would access the same activity across multiple days. We use Chromebooks with school Google accounts, so linking their accounts to Desmos took 2 seconds and was done earlier in the year. I used pacing to restrict them to the section of the day:

This was one of those features I knew was going to come through, but didn't totally trust until I saw it in action. There wasn't a single technical issue over the three days. Each day the Activity Builder remembered the kid had previously accessed the activity and jumped them right to the section of the day. It was really elegant. Sketch slides with a trackpad still kinda stink, but I was not super critical of the results.

Data Use

At the completion of each day, I did some right/wrong (I was pretty unforgiving here) tallying in a spreadsheet, and determined raw scores for the various sections. I also tallied up incorrect answers to see how questions performed. I would eventually throw out the worst performing questions in each section:

After three days of data collection, I set out to determine my final product. What were students going to get about their performance on this giant activity?

The intent of the activity was not to assign a grade based on their raw performance, merely to give them a snapshot of where they stood on December 11-13. Yes, this assessment would factor into their course grade somehow, but I wasn't going to cackle in delight as I failed tons of them, that's not what this was here to do.

Being able to click through dashboard screens and tally results was quicker than I thought, maybe 1 hour a day. Generating something meaningful from the data and formatting it nicely took another couple hours.

The other nice thing about this collection method is I could quickly check for version bias. Each of the three versions had questions that were identical, but others that were modified. Codes were distributed at random, and for whatever reason one version registered a higher average raw score. I curved the other two versions up, roughly 1.16x (normal College Board is 1.20x), so that the group average was the same as the highest average. I took the resulting adjusted score, divided it by max points available, which gave each student a percentage and quartile.

From left to right: class period, fundamentals raw (max 20), curve sketching raw (max 13), skills raw (max 18), raw total (max 48, three questions were deleted), version adjusted total (if required), percentage, quartile. Average was right under 70%. Seven students earned 100%.

I told the students about 10 times, that the percentage was NOT their grade on the assessment, it was merely a tool to see where they landed in the overall population. My message was this is one data point in a series of many and that we would be doing these again. I also wanted to communicate that 1st and 2nd quarter implied you were doing a good job, 3rd quarter meant you needed to study more, and 4th quarter should have been a little wake up call.

After handing out the slips I floated around and had a quick chat with each kid, affirming their work or letting the lower ones know that this number was not a personal judgement, but that something more is required of them.

Conclusion

This went pretty well. The kids took it seriously, the majority of students did well, and I think all of them got useful information out of it. More importantly, this activity was easy to build, easy to manage, and easy to score.

As I've progressed through the career, I have tried to keep track of my base principles. What should always be true about the way I work? What should always be true about the way I run the classroom? And how do you keep it simple to avoid a self-induced pressure cooker?

Know the Content

Above all, I need to know what I'm talking about. I don't want to regurgitate something from a text. I want to make sure I understand a topic, how it works in general, how it might connect to something students have seen before, and how it connects to where I want to go. I want my content to tell a story. It's not necessary that the kids even know they're in the middle of a story, just that they can trust me to talk about things in a logical way that flows nicely. That we don't just study things at random because some curriculum guide told us to do so.

Initially, this was the hard part of the job. I am so upset if I teach something incorrectly, in a tricksy manner, or in an obtuse way. Really grinding away at the content early on has had the biggest payoff. I can sing you the ballad of Pre-Cal complete with a dramatic Third Act in my sleep now. But knowing the content doesn't mean you have to be perfect.

Know the Flaws

I screw up. I admit this to the kids. I make them keep an eye out for my mistakes, because they will happen. I try to model a good attitude when it comes to mistakes. They are ok! Even college educated adults make them! You would not believe the countless mistakes I have made on homework solutions, assessments solutions, and live in the middle of some topic. I recognize that I am going to make mistakes with the math and accept it. I try to minimize them sure, but I don't beat myself up over it. On the off chance I do cover something in a weird or incorrect way, I profusely apologize to the students and make it right.

I also know the flaws of my teaching style. I ramble. I get side tracked. I tell silly stories. The kids know this and in some cases are good at purposely triggering me into a distraction. I have gotten better at recognizing this in the moment and try to minimize the distraction. I don't stop, it makes class fun. It gets the kids to open up and usually leads to each class developing something funny that's uniquely theirs. I have classes that happily sing happy birthday to each and every office aide that wanders in, and that's fantastic.

I misinterpret kids questions and give answers they didn't ask for. I ask questions they don't understand. I think kids are talking to me or about me when that's not the case all the time. I trip over my words. I do all kinds of silly imperfect things. But that's cool, everyone does. It's ok to be a real person in front of students.

Know the People

In 7th grade for whatever reason I approached my art teacher and said very boldly "do you even know my name?" In a true pro move she smiled and said "Jonathan we need to talk about a drawing I want you to make for a contest..." not only deflecting my sorta rude question, but showing me that "ok punk, not only do I know your name, I know you're talented too."

The school I attended in 6th grade had been larger and I felt lost in my big classes (each around 30+). I suppose it was natural to think this teacher generally didn't know me much like the others. And for whatever reason this incident has stuck with me for 20 years. I greatly appreciated all of my teachers who took a moment to acknowledge, yes kid, I know who you are and what you bring to the table.

That's probably the biggest of my base principles. I need to be tight when it comes to presenting and teaching, but I need to be tight on my soft skills too. Each kid should feel like it's ok to talk to me, that we can have a conversation, however brief, that it can be about whatever, and that they know I'm aware of what they're up to and how I might improve things. I have structured so many of my classroom procedures out of building in time for me to get to know the students. If I'm talking 45 minutes a day, every day of the week, that can't happen as well as when I hand out some classwork, turn on the music, and go wandering.

Year Four of Varsity Math. Every year there seem to be weird things that make each group unique, adding new features to our brand. Last year saw the juggernaut of Baby Shark. And boy is that sucker still going strong. We also constructed a Hall of Fame. What's new this year?

Varsity Math has become a brand unto itself and brands have to be managed. You have to keep them in the public eye. The hardest part for us is that 99% of the members graduate. This year we had 0 returning students. For 2018-19 there will be 3. Summer Camp has served as a great on boarding tool, bringing kids into the program in a fun way and hopefully making them that much more excited for the start of the school year. But what about during the school year? How do you get all the kids who didn't go to summer camp and who might only vaguely remember seeing goofy dorks with t-shirts running around?

Promote the crap out of it. Collecting money and generating merchandise takes some time, but once it all arrives I like to have a Nerd Day. A couple weeks ago was the 2017 edition. All the AP kids wear their shirts and patches and stickers on the same day. It gets people talking when ~100 kids not on a sports team all dress alike. At the kids' request I diversified the merchandise and added custom sunglasses:

And in what is by far the goofiest stunt I've heard of, a bunch of them had a parade of sorts at lunch. I was clueless it happened until after the fact when 10 or so kids ran to my room to tell me what they did. It was pretty simple, they did a lap of the cafeteria, sung a poor unsuspecting kid happy birthday and took a group photo:

The contingent that eats lunch earlier in the day was sorely upset that they missed out.

Recognizing that this is the future and something doesn't happen unless it gets recorded on social media, I bought a Snapchat filter for the day. It was geofenced to the classroom and cafeteria sections of the school and I told kids to post, post, post. They did not disappoint:

And the stats were pretty impressive:

I collected as many of them as I could and assembled a giant collage for posting out on the Hall of Fame (2 of 9 pages shown):

The parade was a little silly and over the top, but a sign of how much fun the kids were having. And were you to hang out in my room, you'd find that "over the top" is kind of the status quo anyway. To have the kids in AP math excited to be a part of it is one thing, but the fact that we can generate buzz around the school at large is so awesome. AP math as the cool kids club, who knew?

In July I spent some time at Desmos HQ driving Eli's Lambo and shooting the breeze with people about how they incorporate Activity Builder and how the Desmos staff see as the role of Activity Builder in the classroom. Two things stuck with me: be thoughtful in AB design, and see how could change the way I assess.

Prior to my visit I had decided to experiment with Activity Builder more. I saw a lot of great work with Pre-Cal kids having to explain their thinking more, and Calculus kids could certainly use the same. The language barriers for making mathematical arguments have been a barrier for my students in the past, and I want to start being more picky about that kind of thing. I also want to push my kids to better understand how the regular Desmos calculator works with regard to restrictions, notation, and such.

In 11 weeks I've done 9 actual, premeditated Desmos activities between Calc AB & BC. There's at least another half dozen instances where they used it to make a project or annotated a pre-made calculator page. Here's a small sample of stuff I tried:

Calc AB

Being a math-based LMS, having access to notation is great. Here I asked AB a series of questions about derivative methods:

Being able to let kids sketch is also nice. For an activity on curve sketching, I provided the first derivative and they had to sketch the original function as well as the second derivative:

I really like slides where something has to be added to a graph, makes it easy to see how much a misconception has propagated. Here I can quickly tell that two students misinterpreted the initial picture has f(x) rather than f'(x). Were they working together? Did this idea manifest in separate parts of the room? I can figure it out fast.

Calc BC

For BC my built activities are part of their regular assessment program. I also incorporate it into their classwork a lot. While doing area between curves and volume, I was able to share a calculator page with them and have them add integrals to it. At regular intervals they complete Activity Builders as closed notes (though collaborative) assessments.

Here they shaded regions of a velocity curve where speed was increasing versus when speed was decreasing:

It's also been easy to adapt free response questions to the format:

Since it's a smaller group we've been able to learn a lot of nitty gritty things about the calculator. How to use folders, make dynamic labels, define variables, etc.

Takeaway

In all cases I make sure the activities are short and sweet, usually less than 10 screens. I consider what the activities let me do that's not possible with paper (for instance, compute nasty integrals). I also make sure the kids get to see the data that gets collected. In BC for example, we always go over their assessment when it's finished. I'll call up the dashboard and scroll through interesting answers or demonstrate common issues. At no point are kids being called out to mock them for getting something wrong, rather we use it an opportunity to discuss what mistake they might have made and how what we can all learn from it. Kind of like a digital "my favorite no" kind of thing.

There are still some scaling issues I'm working on. For the most part paper assessments are still faster for AB given the size (75), but there's potential there. So far it's working great. Kids can use the system well, I get useful information from it, and the technology gets out of the way.

As an avid reader of Teacher Internet™, you may think it's all wild innovation and maker spaces in the math universe. Not so! Every day we all have very uninteresting procedures we have to implement, the real workhorse of any classroom. It's the inescapable part you don't read much about. At some point, you have to put away the VR goggles and 3D printers and give an assignment. In my attempt at College Algebra this year, I do exactly that, a lot.

My audience is ~30 kids split into two classes. That's really small, for me anyway. When there's only 15 people in the room, everyone gets face time for as long as any kid could possibly want.

And some of them take all they can get. Another feature of this audience is they're all extremely capable, but we're all over the place in terms of how long it takes for stuff to click and get assignments done. One kid is over and done with the task at hand in 10 seconds, where another needs 5 minutes. Adding to the challenge is just who is super speedy varies depending on the task. It is a weird environment.

Number one message I send to this group is I don't care at all how long it takes you to do something, I want you to understand what you're doing at the end of the task. If that means we're in 15 different places some times, that's fine. Most of these kids have been fighting the speed thing for years, I think it's time to give them a break.

How do you accommodate such a group? With work time, tons of it. I think I stand and talk in front of this group like 20 minutes a week, tops. The rest of the time they're working on problem sets. It is, by far, incredibly uninteresting. But uninteresting != boring.

Yeah, they work all the time, but the tasks are varied. There's some solving, some listing, some writing, and some graphing depending on what we're doing. And at no point are they working on a task to the point of exhaustion. Enter my go to format this year: 9-6-3.

Take a skill, have them perform it on 9 items, do an extension on 6 of those items, and finish with an analysis task for 3 items. Based on my observations the first few weeks of the year, this format keeps everyone at a similar overall pace despite their varied work speeds. 95% of the kids can complete all of that in the time allotted. Although 5% won't quite finish, they really get the ones they dd because they were given the time to sit and contemplate for a while. There's like 10-15% who finish pretty fast and without any prompting by me will be helping other kids around them.

I think the biggest success in this format is other than the initial "do this for all 9" the remainder of the choices are up to the kid. I'll do similar tasks with "here are 6, pick 4." I'm sure there's something in here about choice theory or something, but the kids put up no protests under this system. The kids take ownership here, managing their time to make sure they can do what's expected. That's the other big component, an inspection always comes and they know it. My inspections aren't a big deal, but it's a little thing that sends a big signal: I want you to try something and I want to see how you tried.

In the end all the kids get in reps, they aren't exhausted, and the ones who need to ask a ton of questions have ample time to ask their questions.

A couple weeks ago I had a little fun with BC on Halloween. I liked the mechanics of tying a math problem to some bigger puzzle. This is no different than those escape box kits you can buy for way too much money.

For the first time we were looking at a week off for Thanksgiving, so I wanted something to do on Friday that would be a nice closure. Most people go test here, but I was not in the mood to take a big to do list home. In AB we had made some good progress of curve sketching. Students knew how to start at an original and sketch the two "lower" functions as well as start at a second derivative and sketch the "upper" functions. We'd talked about critical points of the first and second derivative as well as what they represent.

Setup

In this activity students would work with a partner and be given a random 5th degree polynomial as a starting point. On the back was a word, seemingly random.

I started with 5th degree polynomials because it offered a lot of variety. I mixed up the types of behaviors, repeated x-intercepts, and included some with their reflections to ensure students took the time to analyze the functions as we had done. The students objective was to start with their randomly assigned original, find its first derivative from a bank of choices, and then find the second derivative from another bank of choices. All their cards would have a word on the back.

The hallway set up like this:

Minimums and maximums were marked on the f cards. Minimums, maximums, and x-intercepts were marked on the f' cards. X-intercepts were marked on the f'' cards. As I made a lot of similar functions on purpose, I wanted some anchor to help students determine if they were on the right track. They'd have to find the functions with the appropriately matching critical points as well as behavior in order to complete their set.

My largest AB class has 28 students, so there were 14 sets. SOURCE GRAPHS

In Action

How would they know if they had completed their set?

The words on the back of the cards are seemingly random, but they aren't at all. Some time ago I watched a video on the Remote Associates Test (RAT), a mechanism for studying intuition. When presented with three words, they trigger a related fourth word as a response. Some fourth words are more difficult to determine than others. Sets that comply with the RAT methodology are available right here. Students wishing to validate their set would read me the three words they collected. If they formed a valid RAT item, I proceeded to let them guess the fourth word. If one of the words was incorrect, I would tell them which one.

For example, if a student told me ACTOR, FALLING, DUST they had a valid set. The hurdle to their prize was to correctly guess STAR from the prize wall.

Now in an official RAT test, it's not a multiple choice thing. I wasn't really out to do any kind of behavioral study here, so all the 14 possible final words were available for them to see. To prevent random guessing, students had two chances to guess their fourth word. Failure to do so and they forfeit the prize. If they successfully guessed the fourth word, they could open the locker and retrieve the candy inside.

Results

Students would take pictures of their original and walk between the f' and f'' sets while having discussions. Some would retreat to the classroom and sketch the set and go hunting for something that matched. Some theorized they could try to reproduce the function in Desmos, though most abandoned that idea when they realized there were faster methods. The idea was for them have a discussion about function behavior and how f, f', and f'' are linked by critical points.

All students were able to claim their prize, some in as little as 10 minutes. The longest anyone took was about 20 minutes. Most groups produced a valid set on their first try, those failing to do so usually had the second derivative incorrect. In a handful of cases students made a mistake at the first derivative but found the corresponding (though overall wrong) second derivative, meaning their 2nd and 3rd words, while matching each other, were not valid when grouped with their 1st word. One poor kid had the set right the entire time, but had misremembered the first word, leading me to intervene to figure out what the heck happened.

As a point of comparison, I let BC loose on the task too (mostly because I over bought candy and needed a way to get rid of it). They did curve sketching a solid month ago so I was curious to see what they remembered. Unsurprisingly, all of them cracked it in 5 minutes or less.

Conclusion

This passed with flying colors. Wrapping my head around the decoding scheme took some time, but the use of RAT was really clutch here, a ready made puzzle that provided just enough of a pause point to add a nice challenge at the end. Some students spent many many minutes deliberating what their fourth word was, dancing around it the whole time. The two guess rule really put the pressure on. Designing functions that were similar but not too similar took the most time, as well as all the screenshotting, printing, and cutting. Oh, and a lot of nervous labeling. One misplaced word could derail the whole thing, so would losing the sets I used. I clutched that sucker TIGHT.

Loved it though. I'm not sure I've had a first time task go so smoothly.

I forget exactly when, but some BC kid suggested we trick or treat on Halloween. I offered the goofy suggestion of hiding candy in the lockers, making them bring candy buckets, and going door to door.

What we actually did was pretty goofy.

I used 8 lockers, and inside was either a trick or a treat.

I constructed a little puzzle. I borrowed some locks from athletics which have serial numbers. Bulk combination locks have the serial number and a key slot on the back so you can recover the combination or manually open the lock if a kid forgets the way in (or if a random lock appears). The task would have two parts: decode the combination, and find the lock it corresponded to.

For the clues, I had this idea that the lock would be "worth its weight in gold." The serial number would represent a weight, in grams, of gold. If you translate that into moles you get a cryptic looking number. I used that molar count of gold as the envelope label. Inside were three problems, either two derivatives and an integral or two integrals and a derivative. Inside was also a clue mentioning that "two like expressions wouldn't be caught dead next to one another." The intent was kids would realize the combination went D I D or I D I. Evaluating the three problems would yield three numbers that opened the lock in some order. Multiplying the number on the envelope by the g/mol of gold would convert it to the serial number.

I sent out some cryptic hints on our class Remind.

It worked pretty well. They were really confused/interested as to what the heck might be happening. On Halloween, I read some brief instructions, turned off the lights, and they went scrambling. They picked a clue at random and a partner. Five lockers had candy, three did not, but they'd have no idea until cracked the puzzle.

Admittedly I was a little too clever. They figured out the numbers were the combination. Translating the number on the envelope into a serial number had them chasing a lot of random ideas and eventually I had to offer some hints. Most chose to brute force the combination (trying the numbers in various orders until it worked) rather than think about the clue about like expressions. One kid wanted to try all possible versions of his combination on all the locks, a process he didn't realize would take forever.

Overall it worked. The idea just needs a little more refinement. The lock opening phase was particularly intense once they realized it might have all been for nothing.

A follow up from last week. You may have heard of Nix The Tricks, a book that started as a community effort to identify common pitfalls in math concepts.

Tina, compiler and grand Trick Nixer, took notice:

And so we'll see if that idea has some legs. I know the Calculus community is vast, and I have no idea what concepts/progressions need some love. I know what I think is weird, but you, the more experienced Calculus teacher probably have some input.

A Calculus goal was to be more upfront about where some of our patterns come from this year. I really disliked Named Theorems. For example, I have found no purpose if having kids learn the name "Fundamental Theorem of Calculus." It becomes a parrot moment. You say "what's the Fundamental Theorem of Calculus?" and I'm sure they can rattle off the pattern. But is there any connection to what that actually does? Does it improve their ability to calculate a definite integral with that name? I think it's debatable. Talking about final minus initial position has served me better.

Shoutout to my time at the Desmos Fellowship for the following revelations. Sam and Sarah blew my mind with the area model for a derivative of two products. The proof is so beautiful and perfectly suited for high school kids. WAY better than the awful derivation from the limit definition that flew over my head. I used it during Hurricane School with great effect. The question was prompted, well what about the derivative of a quotient? Does that have a proof? I didn't have an immediate answer, but YouTube to the rescue. The relationship made at the beginning was brilliant.

Here are both proofs written out:

Just so simple. And easily discussed in a few minutes. AB got to the product and quotient cases some time later, and I made a point to show them where the patterns come from. I wasn't super interested in them regurgitating the proof, I just wanted them to follow the logic. We were able to have some great discussions about the communicative nature of the two setups. The terms in a product derivative can go in whatever order we choose. A quotient requires us to be more careful. Later we saw that the product can scale to more than two functions.

I decided to cover all our derivative use cases now, rather than coming back to them. Next was exponentials and logs. Twitter to the rescue:

Once you establish e^x is its own derivative, you can show the logic for a natural log:

Again, so great and simple. Props to me for having the foresight to cover implicit derivatives (and isolating dy/dx terms) prior to this, so the dy/dx logic wasn't crazy bananas to them.

In the end, yes, most kids will just learn the patterns as a matter of faith. They will forget these proofs and will probably never reproduce them unless they grow up to be math teachers. But it was important to me to show them that math isn't magic, that we can reason our way to new ideas. Don't take the product rule on faith because I said so.

I can tell you as a student seeing something like this would've been so helpful. Calculus, despite having a great teacher, was a series of rules I felt I had to learn for the sake of learning them. It was many years before I realized how it all fit together.

My College Algebra students can also benefit from my project structures. I have more flexible goals with this group of students, and a main feature is letting them have as much class time as possible to get work done. I set up very brief lessons and let them spend time working. Most of the material is not new and students in here could use more reps.

We spent the first month of the school year talking about linear systems, quadratic systems, and radical equations. In all cases my goal was to show them the importance of a graph and how it related to work they do by hand. Students were to create 3 problems for a set of 4 possibilities: a linear system, a quadratic system with real solutions, a quadratic system with non-real solutions, and a radical equation. In all cases they stated the problem, did the work by hand, and graphed the equation to prove their work. Then they explained their process.

Students were able to use previous classwork as a starting point if they weren't sure how to make up a problem. For most of them they had rarely, if ever, been asked to do something like this. As we have progressed, students have been persevering through their work because checking themselves is so accessible. They don't need me to share an answer key, they have the ability to do it on their own.

As with Pre-Cal and Calculus, I got a lot of variety in the kind of work students turned in and all of them had great conversations along the way thinking about how to represent the situations they chose.